Abstract
In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre (Comm. Pure Appl. Math. 62, 597–638, 2009) are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Hölder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels K σ,β satisfying
with respect to σ ∈ (0, 2) close to 2 (for a given \(\beta \in \mathbb R\)), where the regularity estimates do not blow up as the order σ ∈ (0, 2) tends to 2.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Awatif, S.: Équations d’Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité. Comm. Partial Differential Equations 16, 1057–1074 (1991)
Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 567–585 (2008)
Bingham, N.H., Goldie, C.M., Teugels, J. L.: Regular variation. Cambridge University Press, Cambridge (1987)
Caffarelli, L., Cabré, X.: Fully nonlinear elliptic equations, vol. 43. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence (1995)
Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62, 597–638 (2009)
Caffarelli, L., Silvestre, L.: The Evans-Krylov theorem for non local fully non linear equations. Ann. Math. 2, 1163–1187 (2011)
Di Neza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Jensen, R.: The Maximum Principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rat. Mech. Anal. 101, 1–27 (1988)
Kassmann, M., Mimica, A.: Intrinsic scaling properties for nonlocal operators. arXiv:1310.5371
Kim, P., Song, R., Vondraĉek, Z.: Potential theory of subordinate Brownian motions revisited. arXiv:1102.1369
Kim, Y.C., Lee, K.A.: Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels. Manuscripta Math. 139, 291–319 (2012)
Kim, Y.C., Lee, K.A.: Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels: Subcritical Case. Potential Analysis 38, 433–455 (2013)
Kim, Y.C., Lee, K.A.: Regularity results for fully nonlinear parabolic integro-differential operators. Math. Ann. 357, 1541–1576 (2013)
Krylov, N.V., Safonov, M.V.: An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245, 18–20 (1979)
Lara, H.C., Dávila, G.: Regularity for solutions of nonlocal, nonsymmetric equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 833–859 (2012)
Lara, H.C., Dávila, G.: Regularity for solutions of non local parabolic equations. Calc. Var. Partial Differential Equations 49, 139–172 (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, S., Kim, YC. & Lee, KA. Regularity for Fully Nonlinear Integro-differential Operators with Regularly Varying Kernels. Potential Anal 44, 673–705 (2016). https://doi.org/10.1007/s11118-015-9525-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-015-9525-y